The number of ramified coverings of the sphere by the double torus, and a general form for higher genera

An explicit expression is obtained for the generating series for the number of ramified coverings of the sphere by the double torus, with elementary branch points and prescribed ramification type over infinity. Thus we are able to prove a conjecture of Graber and Pandharipande, giving a linear recurrence equation for the number of these coverings with no ramification over infinity. The general form of the series is conjectured for the number of these coverings by a surface of arbitrary genus that is at least two.Comment: 14pp.; revised version has two additional results in Section

A proof of a conjecture for the number of ramified coverings of the sphere by the torus

An explicit expression is obtained for the generating series for the number of ramified coverings of the sphere by the torus, with elementary branch points and prescribed ramification type over infinity. This proves a conjecture of Goulden, Jackson and Vainshtein for the explicit number of such coverings.Comment: 10 page

Transitive factorizations of permutations and geometry

We give an account of our work on transitive factorizations of permutations. The work has had impact upon other areas of mathematics such as the enumeration of graph embeddings, random matrices, branched covers, and the moduli spaces of curves. Aspects of these seemingly unrelated areas are seen to be related in a unifying view from the perspective of algebraic combinatorics. At several points this work has intertwined with Richard Stanley's in significant ways.Comment: 12 pages, dedicated to Richard Stanley on the occasion of his 70th birthda

Lax matrices for Yang-Baxter maps

It is shown that for a certain class of Yang-Baxter maps (or set-theoretical solutions to the quantum Yang-Baxter equation) the Lax representation can be derived straight from the map itself. A similar phenomenon for 3D consistent equations on quad-graphs has been recently discovered by A. Bobenko and one of the authors, and by F. Nijhoff

Universal factorized formula for the cross-section of two-particle scattering

We analyze the process of two-particle scattering with unstable particle in an intermediate state. It was shown that the cross-section can be represented in the universal factorized form for an arbitrary set of particles. Phenomenological analysis of factorization effect is fulfilled.Comment: 8 pages, corrected typos. change conten

The Parkes quarter-Jansky flat-spectrum sample 3. Space density and evolution of QSOs

We analyze the Parkes quarter-Jansky flat-spectrum sample of QSOs in terms of space density, including the redshift distribution, the radio luminosity function, and the evidence for a redshift cutoff. With regard to the luminosity function, we note the strong evolution in space density from the present day to epochs corresponding to redshifts ~ 1. We draw attention to a selection effect due to spread in spectral shape that may have misled other investigators to consider the apparent similarities in shape of luminosity functions in different redshift shells as evidence for luminosity evolution. To examine the evolution at redshifts beyond 3, we develop a model-independent method based on the V_max test using each object to predict expectation densities beyond z=3. With this we show that a diminution in space density at z > 3 is present at a significance level >4 sigma. We identify a severe bias in such determinations from using flux-density measurements at epochs significantly later than that of the finding survey. The form of the diminution is estimated, and is shown to be very similar to that found for QSOs selected in X-ray and optical wavebands. The diminution is also compared with the current estimates of star-formation evolution, with less conclusive results. In summary we suggest that the reionization epoch is little influenced by powerful flat-spectrum QSOs, and that dust obscuration does not play a major role in our view of the QSO population selected at radio, optical or X-ray wavelengths.Comment: 18 pages, 11 figures, accepted 18 Dec 2004, Astron. & Astrophys. The accepted version is expanded to include an analysis of the form of the decline in radio-QSO space density at high redshifts. This is compared with the forms of epoch dependence derived for optically-selected QSOs, for X-ray-selected QSOs, and for star formation rat

Facing up to the challenge of behavioural observation in infant hearing assessment

The ability to assess detection and discrimination of speech by infants has proved elusive. Dr Iain Jackson and colleagues discuss how new technologies and fresh approaches might offer valuable insight into young infants’ behavioural responses to sound